Deconvolution by digital filtering from linear discriminate analysis

ABSTRACT

A quantitative optical microscopy arrangement is described. Specifically, a digital filter derived from linear discriminant analysis is described for recovering impulse responses in applications that may include photon counting from a high speed photodetector and applied to remove ringing distortions from impedance mismatch in multiphoton fluorescence microscopy. Training of the digital filter is achieved by defining temporally coincident and non-coincident transients and identifying the projection within filter-space that best separates the two classes. The training allows rapid data analysis by digital filtering. The LDA filter is also capable of recovering deconvolved impulses for single photon counting from highly distorted ringing waveforms from an impedance mismatched photomultiplier tube. The LDA filter is also successful in removing these ringing distortions from two-photon excited fluorescence micrographs and may extend the dynamic range of photon counting by about three orders of magnitude through minimization of detector paralysis.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit under 35 U.S.C. § 119(e) of U.S. provisional patent application No. 61/876,743 filed on Sep. 11, 2013, and is a continuation-in-part of U.S. non-provisional application Ser. No. 13/825,341 filed on Jun. 20, 2013, which is a U.S. national application under 35 U.S.C. § 371(b) of international patent application number PCT/US11/53395 filed Sep. 27, 2011, which claims the benefit of priority of provisional applications 61/386,735, filed on Sep. 27, 2010 and 61/457,946, filed on Jul. 14, 2011, the contents of which are hereby incorporated by reference in their entirety into this disclosure.

STATEMENT OF GOVERNMENT INTEREST

This invention was made with government support under GM088499 awarded by the National Institutes of Health, GM103401 awarded by the National Institutes of Health, and GM103910 awarded by the National Institutes of Health. The government has certain rights in the invention.

TECHNICAL FIELD

This application may relate to the detection of particles using detection of particles using detectors with statistical properties.

BACKGROUND

This section introduces aspects that may help facilitate a better understanding of the disclosure. Accordingly, these statements are to be read in this light and are not to be understood as admissions about what is or is not prior art.

Photon counting is a well-known method detecting low intensity light. However, existing approaches for photon counting suffer from nonlinearities at high photon count rates. Several strategies have been adopted for improving the linear dynamic range (LOR) of light detection using photomultiplier tube (PMT) and avalanche photodiode detectors (APD), or other detector types having similar statistic properties.

For example, neutral density filters can attenuate high light levels so as to remain within the linear range of counting systems. The response time of this technique is not fast enough to provide large continuous dynamic ranges in rapid sampling applications and requires careful calibration of optical density for the filters. Another method uses a plurality of photo detectors and fiber-optic beam splitters to sample different fractions of the beam, equivalent to performing simultaneous photon counting with several neutral density filters. Using this approach, 6 orders of magnitude of linear been response has been achieved.

Other methods include the combining of photon-counting detection for low-light levels with analog-to-digital conversion (ADC) so as to extend the LOR to the high photon flux regime. Another method includes fast ADC of the temporal time-trace followed by Fourier transformation to deconvolve the number of photons present in a time window, but this requires long analysis times and fast ADC (˜1 GHz sampling).

Each of these approaches involves performance trade-offs. Detectors optimized for photon counting with fast rise/fall times are generally not optimized for ADC and vice versa. Sensitivity mismatch in the instrument responses from single photon counting (SPC) with ADC may impact reliable quantitation and may require simultaneous data acquisition using two fundamentally different electronics approaches. The noise contribution from combinations of multiple detectors is additive. In addition, differences in sensitivity and drift may compromise the accuracy when stitching together the results from multiple detectors.

The relationship between the detected count rate and the selection of the threshold voltage of a counting discriminator(s) has been studied. Use of multiple thresholds to improve the dynamic range of photon counting systems from a single-channel detector has been demonstrated In measurements with pulsed excitation and long times between pulses relative to the detector response time (e.g., multi-photon and nonlinear optical microscopy at <100 MHz laser pulse repetition rates with detector fall times <10 ns); the voltage transients from the single photon events can be reliably treated as temporally coincident. Detection of up to 4 simultaneous photons per laser pulse was achieved by careful adjustment of the detection voltage threshold of each discriminator to fall between the peak voltage distributions of n and n+1 simultaneous photons. This approach suffers in practice from the relatively large intrinsic variations in peak voltage distributions for a single photon in most practical photomultiplier tubes. Since the mean and variance in the peak voltage distribution increase linearly with the number of photons, the distributions for and n and n+1 photons quickly overlap as n increases, rapidly increasing the uncertainty in attempts to quantify the number of simultaneous photons with this approach.

SUMMARY

Presented herein is a quantitative optical microscopy arrangement, which has a light source configured to provide optical energy to a sample region, and a detector configured to receive light reflected from or emitted through the sample. The detector has an optical-voltaic converter configured to convert the received light into a representative voltage, and a digital deconvolution filter configured to deconvolve impulse response of the detector and signals associated with photons. The digital deconvolution filter includes a plurality of coefficients determined based on a linear discriminant analysis.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1A, shows representative calculated lognormal voltage distributions for n photon events based on the same initial parameters, M_(L,1) and S_(L,1)·M_(L,n) and S_(L,n) calculated using equations (10) and (11); and FIG. 1B, the CPHD for various mean number of photons per trial. Each curve integrates top instead of 1 because no voltage is generated for zero photons.

FIG. 2 shows an example relating Poisson distribution of photons and binomial probability, p.

FIG. 3 shows the signal to noise ratios (μ/σ) calculated using equations (15) and (16) and theoretical maximum based on the Poisson distribution are compared for N=100,000 trials. The vertical bars mark the p values that correspond to given means, and the multiplier indicates the increase in noise relative to the theoretical limit.

FIG. 4 shows the value of p for different threshold settings as a function of the mean number of photons. Different thresholds have sensitivity to different photon ranges, and the overlap in these curves results in continuous quantitation.

FIG. 5 is a representation of second harmonic generation optical equipment configuration.

FIG. 6 shows peak voltage distributions for two incident photon fluxes. The solid lines are the best fit theoretical Poisson weighted sum of lognormal distributions with parameters determined from single photon voltage distributions. The green points and pink line are rescaled by a factor of ten for visualization.

FIG. 7 shows a comparison of observed counts (triangles) and calculated photons (diamonds) outside of the Poisson counting range (λ>0.1 03) as calculated by the single threshold method in equation (15).

FIG. 8 shows a comparison of conventional SPC (triangles), signal-averaging ADC (squares) and calculated photons (diamonds) using multiple counting threshold settings. The ADC measurements were rescaled to overlay to account for the proportionality between photons and voltage.

FIGS. 9A-9B show histograms of voltage peak distributions measured at single pixel locations in an image, where 9A is a pixel with no signal counts and dominated by Johnson noise, and, 9B, is a pixel with sparse counts. The two images are shown on different voltage scales.

FIGS. 10A-10C show images resulting from statistical analysis of the peak height distributions of raster scan microscopy data, where FIG. 10A, is photon counting image with a threshold selected based on the measured one-photon peak height distribution and the Johnson noise. Each pixel has units of counts measured by discrimination; FIG. 10B, is a signal-averaged image, in which each pixel is the mean detector voltage; and FIG. 10C, a stitched image with the signal-to-noise optimized at each pixel using either the discrimination or the mean values (the bottom image is comprised of the mean of the Poisson distribution reported in each pixel, with a continuous dynamic range spanning a mean of 0.002 photons per pulse to a mean of 75 photons per pulse (i.e., spanning the low-light limit set by dark counts up to 5.9 billion photons per second in this case).

FIG. 11 graphically depicts temporally coincident and non-coincident signals (top). After linear discriminant analysis, a digital filter is constructed, along with projects of the temporally coincident and non-coincident signals.

FIG. 12 shows the simulated detector data of three measured intensities.

FIG. 13 is a receiver operating characteristic curve for LDA filtered data.

FIG. 14a is a voltage versus time graph of deconvolution of the experimentally measured waveform with ringing to recover an impulse response, specifically a representative waveform produced upon single photon absorption.

FIG. 14b is a voltage versus time graph of deconvolution of the experimentally measured waveform with ringing to recover an impulse response, specifically a filter produced by LDA in order to perform the deconvolution.

FIG. 14c is a voltage versus time graph of deconvolution of the experimentally measured waveform with ringing to recover an impulse response, and specifically representative measurements prior to filtering, filtering with the LDA-based filter, and filtering with both an LDA and High Pass (HP) filter.

FIG. 15 shows deconvolution of the ringing waveforms with pulse pileup.

FIG. 16 shows a comparison of different count methods for ringing data.

DETAILED DESCRIPTION

Exemplary embodiments may be better understood with reference to the drawings, but these embodiments are not intended to be of a limiting nature. In the following description, specific details are set forth in order to provide a thorough understanding of the present invention which, however, may be practiced without some or all of these specific details. In other instances, well known process operations have not been described in detail in order not to unnecessarily obscure the description.

It will be appreciated that the methods described and the apparatus shown in the figures may be configured or embodied in machine-executable instructions, e.g. software, or in hardware, or in a combination of both. The instructions can be used to cause a general purpose computer, a special-purpose processor, such as a DSP or array processor, or the like, that is programmed with the instructions to perform the operations described. Alternatively, the operations might be performed by specific hardware components that contain hardwired logic or firmware instructions for performing the operations described, or by any combination of programmed computer components and custom hardware components, which may include analog circuits.

The methods may be provided, at least in part, as a computer program product that may include a non-transient machine-readable medium having stored thereon instructions which may be used to program a computer (or other electronic devices) to perform the methods. For the purposes of this specification, the terms “machine-readable medium” shall be taken to include any medium that is capable of storing a sequence of instructions or data for execution by a computing machine or special purpose hardware and that cause the machine or special purpose hardware to perform any one of the methodologies or functions of the present invention. The term “machine-readable medium” shall accordingly be taken include, but not be limited to, solid-state memories, optical and magnetic disks, magnetic memories and optical memories. The description of a method as being performed by a computer should not preclude a portion of the same method being performed by a person.

For example, but not by way of limitation, a machine readable medium may include read-only memory (ROM); random access memory (RAM) of all types (e.g., S-RAM, D-RAM. P-RAM); programmable read only memory (PROM); electronically alterable read only memory (EPROM); magnetic random access memory; magnetic disk storage media; and, flash memory.

Furthermore, it is common in the art to speak of software, in one form or another (e.g., program, procedure, process, application, module, algorithm or logic), as taking an action or causing a result. Such expressions are merely a convenient way of saying that execution of the software by a computer or equivalent device causes the processor of the computer or the equivalent device to perform an action or a produce a result, as is well known by persons skilled in the art.

When describing a particular example, the example may include a particular feature, structure, or characteristic, but every example may not necessarily include the particular feature, structure or characteristic. This should not be taken as a suggestion or implication that the features, structure or characteristics of two or more examples should not or could not be combined, except when such a combination is explicitly excluded. When a particular feature, structure, or characteristic is described in connection with an example, a person skilled in the art may give effect to such feature, structure or characteristic in connection with other examples, whether or not explicitly described.

An analytical expression is described that may be used in an apparatus and method to extend the linear range of traditional (single threshold) photon counting measurements to ˜74 simultaneous photons, and a multi-threshold detection method is described that may extend LDR range up to that of ADC signal averaging. The inability of voltage discrimination to reliably distinguish between single photon and simultaneous multiple photon events, which was limiting in the previous techniques is overcome by the technique described herein.

After determining the expected single-photon peak-height distribution, convolved peak-height distributions for multiple photon events are calculated, and then summed by a Poisson weighting to generate the net collective peak height distribution (CPHD) for an assumed mean of a Poisson distribution. The mean number of counts expected for a given discriminator threshold is calculated from the CPHD and compared with experimentally measured counts exceeding the threshold. Values for the mean of the Poisson distribution are then iteratively adjusted by a single parameter weighted least squares minimization.

An objective of quantitative photon counting measurements may be the accurate determination of the mean of the underlying Poisson distribution, ƒ_(P).

$\begin{matrix} {{f_{P}x} = {\frac{\lambda^{x}}{x!}e^{- \lambda}}} & (1) \\ {\mu_{P} = {{\lambda\mspace{31mu}\sigma_{P}^{2}} = \lambda}} & (2) \end{matrix}$

However, the use of discriminators in the detection process yields two possible outcomes: either the voltage transient exceeds a threshold or it does not. Therefore, the output of conventional photon counting instrumentation is described by a binomial probability density function (pdf). Here, the probability ƒ_(B) of observing a particular number of counts x for a given discriminator threshold in a photon counting experiment follows the binomial distribution.

$\begin{matrix} {{f_{B}x} = {{\frac{N!}{{{x!}\mspace{20mu} N} - {x!}}p^{x}\mspace{14mu} 1} - p^{x}}} & (3) \\ {\mu_{B} = {{N_{P}\mspace{31mu}\sigma_{B}^{2}} = {{N_{P}\mspace{14mu} 1} - p}}} & (4) \end{matrix}$ where μ is the mean and σ is the standard deviation.

In (3), p is the probability of a successful outcome (e.g., a voltage exceeding a discriminator threshold), and N is the number of measurements. In the limit of low values of p and high values of N binomial distribution converges to the Poisson, allowing μ_(B) (e.g. the measured photon counts) to be used as μ_(P). However, these two distributions diverge as μ_(P) increases and the number of events in which two or more photons are generated per laser pulse becomes significant. Most conventional photon counting approaches are restricted to this low flux regime where the two distributions converge. However, the binomial distribution is still valid for describing experimental measurements with higher count rates.

The method and apparatus described herein permits determining the mean of the underlying Poisson distribution of incident photons from the measured binomially distributed values. This may extend the range over which Poisson-distributed counts can be recorded by photon counting methods. The signal-to-noise ratio (SNR) is maintained close to the theoretical Poisson limit of conventional photon counting for photon arrival rates over nearly the entire range of measurement. The discussion is focused on pulsed-mode operation, such as might arise in multi-photon or nonlinear optical microscopy using high-repetition rate ultrafast laser sources. However, the approach is general to any counting measurement that is based on electron multiplication, or similar effect.

The distribution in detected peak voltages and the number of photons arriving at the detector may be deduced from the nature of the amplification process. Each photon incident on a PMT ejects a single photoelectron from the photocathode. That ejected electron is ejected towards the first dynode. When the electron strikes the first dynode, the electron imparts sufficient energy to release several more electrons which are accelerated towards the second dynode. This process continues until a large burst of electrons reaches the anode, generating current transients large enough to be discriminated from the background noise. At each dynode there is a random amplification factor or gain that describes the probability of a given number of electrons being ejected for each incident accelerated electron from the preceding dynode.

The total gain per initial photoelectron (V) is a product of the gain at each dynode. Since the gain at each dynode is an independent, random number, the pdf of V converges to a lognormal distribution in accordance with the Central Limit Theorem for multiplicative processes.

The log-normal peak voltage distribution for a single initial electron, ƒ_(L,1) with scale and shape parameters M_(L,1) and S_(L,1), generated by an electron multiplying device may be expressed as:

$\begin{matrix} {{f_{L,1}V} = {\frac{1}{\overset{\_}{2\pi}\; S_{L,1}V}e^{\frac{- {({{lnV} - M_{L,1}})}^{2}}{2\pi\; S_{L,1}^{2}}}}} & (5) \\ {\mu_{L,1} = e^{M_{L,1} + {S_{L,1}^{2}2}}} & (6) \\ {\sigma_{L,1}^{2} = {{e^{{2M_{L,1}} + S_{L,1}^{2}}\mspace{14mu} e^{S_{L,1}}} - 1}} & (7) \end{matrix}$

In the event photons, n where n>=2), arrive at the detector simultaneously, the peak voltage generated will be a sum of n random numbers from the single photon voltage distribution. The pdf for an n photon event is given by n−1 convolutions of the single photon response with itself. f _(L,n) =f _(L,1)

f _(L,1) ^(n−1)  (8)

Although no analytical form is known for even the first convolution of two lognormal pdfs, the resulting pdfs from the convolutions can often be remarkably well-approximated as lognormal under the conditions typical of photon/electron counting. Using the expressions for the propagated mean and variance of the sum of n statistically independent random numbers, μ_(L,n) =nμ _(L,1)σ_(L,n) ² =nσ _(L,1) ²  (9) and the expressions given in (8) for a lognormal pdf, rearrangement of the equations yields the following expressions for the shape parameters M_(L,n) and S_(L,n) for the lognormal pdf after n−1 convolutions:

$\begin{matrix} {M_{L,n} = {\ln\frac{{n^{3}e^{2M_{L,1}}} + S_{L,1}^{2}}{n - e^{S_{L,1}^{2}} - 1}}} & (10) \\ {S_{L,n} = {\ln\frac{n + e^{S_{L,1}^{2}} - 1}{n}}} & (11) \end{matrix}$

The theoretical pdfs for multiple photon events are shown in FIG. 1 using representative values for the mean and standard deviation of the single-photon lognormal distribution.

Taking account of both the Poisson distribution of photons and the log-normally distributed peak voltages associated with a number of photons, n, incident on the PMT, the value of p (i.e., the probability of a successful observation of a count) in (3) can be expressed, for any threshold, as:

$\begin{matrix} {p_{threshold} = {\sum\limits_{n = 1}^{\infty}{\frac{\lambda^{n}}{n!}e^{- \lambda}{\int_{threshold}^{\infty}{\frac{1}{\sqrt{2\pi}S_{L,n}V}\ e^{\frac{- {({{\ln{(V)}} - M_{L,n}})}^{2}}{2\pi\; S_{L,n}^{2}}}{dV}}}}}} & (12) \end{matrix}$

In practice, the threshold in the above expression is adjusted as low as possible such that the integral for any n goes to 1 (i.e., approaching the limit in which every time one or more photon initiates an electron cascade), every resulting voltage transient is expected to exceed the threshold. This theoretical limit may be closely approximated experimentally with appropriate detectors, and data processing. In this limit, evaluation of the integral is no longer required, leaving only the Poisson probabilities for n photons. Np then increases linearly until the Poisson probability for n>1 becomes significant. When λ≈0.103 (p=0.098) or higher, the binomial observable, Np, will underestimate the true number of photons by about 5%, representing the practical upper limit of the linear range for conventional photon counting. FIG. 2 demonstrates that the observed counts are lower than the true number of photons. The binomial parameter p, which in this plot is 0.9, describes the integrated probability of observing one or more photons. As many as about 8 photons may arrive at the detector simultaneously under these conditions, placing the number of counts outside linear photon counting range.

An analytical expression can be derived that relates the counts and photons that takes advantage of the Poisson statistics afforded by simultaneous photon arrival. If the threshold is low enough, the expression for p converges to the Poisson cumulative distribution function (cdf):

$\begin{matrix} {p = {{\sum\limits_{n = 1}^{\infty}{\frac{\lambda^{n}}{n!}e^{- \lambda}}} = {1 - e^{- \lambda}}}} & (13) \end{matrix}$

The expression for the most probable value of λ and the variance in the value in terms of observed counts can be derived using this expression for Np as the fitting function in a weighted one parameter nonlinear fit.

$\begin{matrix} {{x^{2}\lambda} = \frac{{counts} - {N\; 1} - {e^{- \lambda}}^{2}}{{N\frac{counts}{N}1} - \frac{counts}{N}}} & (14) \end{matrix}$ x² is minimized when

$\begin{matrix} {x^{2} = {{{- \ln}\; 1} - \frac{counts}{N}}} & (15) \end{matrix}$

The variance in this most probable value for can be determined from the second derivative of x² evaluated at the minimum.

$\begin{matrix} {\sigma^{2} = {2\frac{d^{2}x^{2}}{d\;\lambda^{2}}}} & (16) \end{matrix}$

While this method may be valid for counting measurements of any Poisson distributed process in the limit in which every initial event (e.g., electron cascade) is guaranteed to generate a detected count, there are two practical considerations. First, any realistic detector will also have dark counts, (e.g., counts generated by spontaneous initiation of an electron cascade). The dark counts may be subtracted from the calculated value of λ, in the low-count limit where they are a significant influence. Second, even though this expression may properly calculate the correct mean number of photons, μ_(p), the variance will be slightly larger than the limiting value of σ²=μ_(p) for values of μ_(p) outside the Poisson counting range. FIG. 3 illustrates the calculated signal to noise ratio (μ/σ) as a function of the calculated f.Jp and a comparison to the ratio intrinsic in the Poisson distribution (i.e., the theoretical limit). The curve tracks closely with the square root of counts for low counts (i.e. within Poisson counting range). Indeed, this ratio is only a factor of 2 lower than the Poisson limit when the probability of observing a count exceeds ˜90% of N.

The highest quantifiable mean number of photons is limited only by N. Depending on the precision required by the specific application it is the N−1 counts may be converted into the estimated number of incident photons. The maximum quantifiable number of photons is given by In N. For example, this method would extend the linear range to ˜11 photons per pulse, on average, if N=100,000 and there are negligible dark counts.

It not always practical to simply increase N so as to extend the linear range, since increasing N by a factor of 10 results in a 10-fold increase in the measurement time with only ln(10)=2.303 increase in dynamic range. A method for further extension of the linear range is to set higher thresholds that will only ever be exceeded by many simultaneous photon events. However, interpreting the results from such higher thresholds includes accounting for the lognormal peak voltage distribution in the expression for p in (13). Specifically, p is given by integration over the pdf of the voltage distribution (p=1−cdf, evaluated at that threshold value).

Representative curves that relate p and photons for various thresholds are shown in FIG. 4, demonstrating the quantitative range for different threshold values. It may be possible to set a plurality of threshold voltages such that the linear dynamic range can be extended all the way to (and perhaps beyond) the range available to signal averaging measurements, limited only by roll-off in the PMT linear response from electron depletion of the dynodes. In this analysis, the same one parameter weighted-nonlinear-fit for the mean of the Poisson distribution is used, except it is evaluated numerically.

In an example an apparatus generating visible light intensities over 8 orders of magnitude is shown in FIG. 5, which depicts the optical setup used to generate these intensities by second harmonic generation (SHG). The apparatus includes a pulsed infrared laser 10, a Glan polarizer 20, 1064 nm zeroth order halfwave plates 30 a visible wavelength blocking filter 40, a KTP (Potassium Titanium Oxide Phosphate (KTiOPO₄) frequency doubling crystal 50, and IR blocking filter 60 and a PMT 70.

A pulsed laser (JDS Nanolaser, 1064 nm, 6 kHz, 0.5 ps pulse duration, 40 mW average power) was directed through two attenuators) before reaching a frequency doubling crystal (ALPHALAS, Goettingen, Germany). The intensity of the light to be detected was controlled by selection of the rotation angles of the zero-order halfwave plates 30 (calibrated to within 0.1°) used in the attenuators. Each attenuator produced 4 orders of magnitude variation in SHG intensity. With the incident light attenuated as much as possible, the angle of incidence of the light on the frequency doubling crystal was adjusted until the observed counts were well within Poisson counting range.

Two different photomultiplier tubes (PMT) were used. A Burle 8850 (Lancaster, Pa.) was used in the confirmation of the log-normal pdf. A Photon is XP2920 (Merignac, France) was used for the extended linear range. Beeker-Hickel counting cards (MSA 1 000) (Berlin, Germany) were used for impulse counting with 1 ns bins, triggered from the laser. For the higher signal intensities, 10 dB attenuators were added to the signal channel, to maintain the signal level within the range of thresholds available from the counting cards. The signal averaging was performed using an oscilloscope (Tektronix 3054B, Beaverton, Wash.). The data analysis (e.g. single and multiple parameter weighted nonlinear fitting) was performed on software written in MathCad 14 (Parametric Technology Corporation, Needham, Mass.) for this purpose using built-in mathematical functions to evaluate Poisson probabilities and log-normal cdf values.

The voltage distributions for various incident photon fluxes were measured and compared with the peak voltage distributions predicted by a weighted sum of related lognormal distributions. Integrals over the voltage distributions were measured by setting successively higher thresholds and measuring the counts. The derivatives with respect to the threshold were calculated to recover the voltage distributions. For the one-photon voltage distribution, a three parameter unweighted nonlinear fit was performed to determine λ, M_(L,1) and S_(L,1). The best fit value for M_(L,1) was −5.025 with a standard deviation of 0.019 and for S_(L,1) was 0.275 with a standard deviation of 0.021. Curve 100 in FIG. 6 represents the voltage distribution within the Poisson counting limit (scaled up by a factor of 10 so as to be better viewed), where the probability of observing two or more simultaneous photons per laser pulse was much less than one.

Setting M_(L,1) and S_(L,1) as constants allowed prediction of the lognormal pdfs for an arbitrary number of photons. Using the mean of the Poisson distribution λ as the only adjustable parameter, fits were generated for λ=0.610 with a standard deviation of 0.029 for the upper set of points (curve 110), and λ=0.01333 with a standard deviation of 0.00050.

Using only the M_(L,1) and S_(L,1) parameters for a given PMT, the mean number of photons were calculated. The observed counts and the calculated photons are compared in FIG. 7 as a function of the incident intensity for the range just outside of Poisson counting range. For N=105, the Poisson limit is 1.03×10⁴ counts (λ=0.103). The highest observed number average number of photons was 4.71×10⁵ (λ=4.71 with a standard deviation of 0.033), corresponding to a nearly 50-fold increase in linear range using the analytical single threshold method described by (16). The signal to noise ratio at this extreme value was ˜5 times lower than the theoretical maximum as estimated by Poisson statistics.

To extend the range further, several higher thresholds were chosen. The counts observed for the higher thresholds were converted to photons by numerically minimizing x²(λ) for each threshold. Only a single parameter was used in the fits, since M_(L,1) and S_(L,1) were determined experimentally under conditions in which the Poisson and binomial distributions converged. In addition, signal-averaging measurements consistent with analog-to-digital conversion (ADC) approaches were performed simultaneously with 512 averages. FIG. 8 shows these results.

The linear range for photon counting may be limited on the low end by the dark count rate and on the high end, as discussed above, by the deviation of the binomial distribution of counts and the Poisson distribution of photons (λ=0.1 03 or p=0.098). For the PMT used in FIGS. 7 and 8, the dark count rate was 7.8×10⁻⁶ counts per laser pulse, yielding a linear range of ˜4 orders of magnitude using conventional photon counting approaches. The linear range for signal averaging was limited on the low end by Johnson noise and on the high end by charge depletion of the dynodes. The signal averaging data were rescaled to overlay with the extrapolated linear fit from the SPC method for comparison. In this experiment, the linear range for signal averaging begins at ˜1 mV (λ=0.045) and extends to ˜890 mV (λ=45) providing ˜3 orders of magnitude of linear range.

The analysis method described thus was has a linear range extending from the dark count rate all the way to saturation of the detector: ˜7 orders of magnitude, spanning the entire range of both SPC and signal averaging. A total of 6 thresholds were used to cover this range. Analysis of the counts from the first threshold provided ˜5.6 orders of magnitude and the other thresholds were used for the next ˜1.4 orders of magnitude. The average standard deviation of the calculated number of photons, above Poisson range and below saturation was 1.5 times the theoretical limit.

Extending the multi-threshold counting approach to provide a continuous linearity throughout the large intrinsic linear dynamic range of a PMT, hundreds of threshold may be used to maintain maximum signal-to-noise ratio over the entire measurement range. Realization of more than hundreds of discrimination thresholds can be achieved in practice by digitization of the peak voltage of each signal pulse using an ADC. This method intrinsically results in hundreds or even thousands of “thresholds”.

The high-resolution voltage discrimination that digitization affords leads to direct recovery of the voltage distributions, simplifying the analysis to reduce the dependence on nonlinear curve fitting with widely spaced discrimination levels. In addition, the voltage distributions for single signal events and for the noise can be continuously monitored in areas with little or no signal, instead of being measured once before at the start of experiments. This allows for self-calibration of the detector response and can provide access to the full dynamic range of the detector using analogous statistical methods to the multi-threshold counting method. An example of peak height distributions measured at single pixels during nonlinear optical imaging is shown in FIG. 9A and FIG. 9B.

The approach uses the ability to digitize the peak of the signal voltage pulse by the ADC by clocking or triggering the ADC such that the voltage is sampled and digitized in time synchronism with the laser pulse. In order to perform these measurements with a detector with a response time of about 1 ns, the bandwidth of the ADC sample and hold circuit should be sufficient to respond to such short pulses. Using a 16-bit ADC, and an 80 MHz or 160 MHz laser repetition rate used in this example, with two channels of detection, the data rate is 640 MB/s. A PCIe or other modern bus can transfer this amount of data to computer memory. For continuous acquisitions, the data is processed or saved to non-volatile memory faster than they are acquired. For the analysis methods described herein, specifically linear fitting, a graphics processing unit (GPU) may be used increase the processing speed for performing real-time data analysis.

In an aspect, a method for using digitized data to perform counting, using an analytical counting linearization relationship, and a signal averaging simultaneously. Counting is performed by comparing each digitized value to a threshold based on the highest voltage expected from the noise. Those same voltages measured for each laser pulse can then be averaged. The counting results are used in areas of low signal and the signal averaging results are used for high signal. The voltage is related to the number of photons incident on a detector by comparing the results for counting and averaging in the regime where the average number of signal generating events per laser pulse is about 1.

This method is self-calibrating because the counting threshold can be adjusted as the noise changes and the relationship between average voltage and number of signal generating events is continuously adjusted in case the detector response characteristics change.

Second harmonic generation microscopy images resulting from this analysis method are shown in FIGS. 10A-10C. These results were acquired with a 160 MHz laser (1000 nm, 100 mw, Spectraphysics MaiTai, Santa Clara, Calif.) whose repetition rate had been doubled. The digitizer was clocked to that repetition rate and 16-bit samples were recorded on two channels simultaneously (AiazarTech ATS9462, Pointe-Claire, QC, Canada). The entire image required about 3 seconds to acquire using a raster scanning technique. Stitching of counting and averaging images in this way requires overlap between the dynamic ranges of detection for both techniques, which was enabled in this case through the use the algorithms given in (12) and (13) connecting the Poisson and binomial distributions for extending the dynamic range of discrimination-based counting.

Since the entire distribution of peak voltages is available in each pixel of the measurement (FIGS. 9A-9B), is possible to perform a linear fit to the voltage distributions of the pixel. The noise and signal voltage distributions are measured in a similar way, as has previously described. After each section of data is acquired, the data can be transferred to a GPU. Linear curve fitting to the measured voltage distributions are then calculated. The weighted mean and variance of the underlying Poisson distribution of counting events per excitation event (in this case, per laser pulse) are calculated for the linear fitting coefficients and may be returned as the results in real time.

The examples provided herein us detection of light generated from pulsed excitation. However, there are applications for this method that do not involve signals timed to a pulsed laser, such as mass spectrometry and many fluorescence measurements. A problem associated with nonpulsed applications is that it can no longer be assured that the signal that is recorded corresponds to the peak of the signal voltage pulse. There may also be the potential of overlap between signal pulses shifted in time. The measured voltage distribution will be complicated by the temporal response of the signal pulse and the temporal overlap such that the relationship between the single event response and the higher number of event responses will be unclear.

Precise knowledge of the temporal response of the detector may complicates algorithms based on linear fitting of the peak height distribution, but still allows data analysis based on connecting the analytical counting linearization and signal averaging. When using the analytical counting approach, the time between digitizations should be greater than twice the rise and fall time of the detector. In this case, no signal will appear on two different digital measurements. The signal averaging lower detection limit is also raised slightly due to the increased variance in the signal. This data analysis method allows the same electronics to be used for collection and analysis pulsed and non-pulsed signals.

Since this method can be used with pulsed signals or nonpulsed signals, it can be applied to many other experiments that use similar types of detectors and can have similar dynamic range issues. For example, mass spectrometers frequently use ion-multiplying tubes to generate short pulsed signals when an ion strikes the face of the detector. This allows for choosing to count ions or average the voltages they generate in an analogous way that photons are detected by PMTs.

When data acquisition is performed with a small number (<20) of discriminator measurements, rather than direct digitization of the individual voltage transients, the approach described herein needs to accommodate practical aspects of the measurement devices. Ringing from slight mismatches in impedance can result in multiple counts from a single event (i.e., after-pulsing). However, time gating and/or careful impedance matching can usually alleviate this issue. At high pulse repetition with high rates, the time between pulses may be relatively short, and the voltage transient may not recover to baseline before the next transient is generated. In this case, thresholds set to low magnitudes may not count the next pulse, biasing the measured counts. This problem may be mitigated, for example, by increasing the time between pulses, reducing the response time of the detector, or correcting the low-threshold data for this effect. Direct digitization largely alleviates complications from after pulsing as well as drift in the DC background and reduces complications from the detector response time.

Gradual drift in the gain (i.e. M_(L,1) and S_(L,1)) can also adversely affect the analysis based on discriminator processing alone. The single threshold method is relatively insensitive to this problem as long as the detector gain does not drop low enough that the probability of a pulse not exceeding the threshold becomes significant. However, the multithreshold method explicitly relies on accurate characterization of the gain distribution and could produce different results if there is a significant discrepancy between the assumed and actual gain. The stability of a PMT can be characterized before applying methods for quantization of voltages. Alternatively, direct digitization provides a route for experimentally measuring the peak-height distribution directly to enable dynamic corrections based on the measured instrument response.

Three related and complementary statistical approaches were demonstrated for extending the linear range in photon counting measurements of coincident photons (e.g., using pulsed laser excitation). A simple expression relating observed counts and mean number of photons was derived to extend the linear counting range past the traditional limit on photon counting. A more general form of this expression based on combined results from multiple-threshold discrimination can be used to numerically extend the linear range to the limit where the PMT voltage response departs from linearity due to electron depletion of the dynodes. Further extension of the multi-threshold measurement to direct digitization of each voltage transient can enable continuous optimization of signal-to-noise ratio throughout the entire intrinsic linear dynamic range of the detector. Direct digitization can also enable real-time measurement of the DC offset, background Johnson noise characteristics, and the detector response function to allow real-time optimization of the statistical analysis.

A method for statistically analyzing the data acquired by using discriminator-based counting electronics to measure a number of signal generating events on a short pulsed detector includes relating the binomially distributed measurement of counts and the Poisson distributed signal of discreet events, using nonlinear curve fitting to a Poisson weighted sum of related detector response functions to relate the binomially distributed measurement of counts and the Poisson distributed signal of discreet events.

In an aspect, a method for measuring a range of voltages from a detector in pulsed signal measurements includes: digitizing the signals from the detector, synchronizing the digitization to the times when the signal is expected, and performing statistical analysis of the distribution of digitized signals to connect the binomial and Poisson distributions to connect the dynamic ranges of counting and signal averaging.

In another aspect, a method for measuring a signal from a single detector in continuous signal measurements includes: digitizing the signals from a short pulse detector and performing statistical analysis of the distribution of digitized signals to connect the binomial and Poisson distributions to connect the dynamic ranges of counting and signal averaging.

In yet another aspect, a method for measuring a range of voltages from a single detector in pulsed signal measurements includes: digitizing the signals from a detector, synchronizing the digitization to the times when the signal is expected, and performing linear fitting of the measured peak height voltage distribution to a linear combination of known functions, comprising the expected peak height distributions of the detector as a function of the number of signal generating events and the electronic noise.

While the methods disclosed herein have been described and shown with reference to particular steps performed in a particular order, it will be understood that these steps may be combined, sub-divided, or reordered to from an equivalent method without departing from the teachings of the present disclosure. Accordingly, unless specifically indicated herein, the order and grouping of steps is not a limitation of the present disclosure.

In addition, integration of a digital deconvolution approach can pre-condition transient signals for extending linear dynamic range. The signal transient from a PMT or a photodiode includes a convolution of the initial photon absorption event with the instrument response function of the detector. As a consequence, the measured transient generated from a single photon impulse event can exhibit temporal broadening and ringing that complicate threshold analyses for dynamic range enhancement. There are many strategies for dealing with problems associated with this convolution, such as nonlinear curve fitting which is routinely done in spectroscopy and chromatography. However, these typical strategies require some prior knowledge of the functional form for the convolution, require initial guess values, and require multiple iterations to achieve convergence such that they are ill-suited for real-time analysis of streaming data.

Among the greatest limitations to performing nonlinear curve fitting is arguably the time required to perform the fit. Examples include demonstrations with streaming data rates at up to about 160 MHz on multiple acquisition channels, leaving just over 6 ns for real-time fitting of a single data channel, which is incompatible with the time required to perform nonlinear fitting given. In principle, deconvolution of the signal and instrument response function can be performed quickly by digital filtering with a filter representing the inverse of the instrument response function. In practice, maximizing the signal to noise ratio in the recovered deconvolved data set involves detailed understanding of both the signal and the noise in the measurement. Most analytical approaches for the generation of digital deconvolution filters require explicit low-noise a priori knowledge of the instrument response function and particular assumptions regarding the probability distribution underlying the measurement noise. In photon counting applications with significant Poisson noise, most standard models are not directly applicable and should be evaluated on a case by case basis.

In an embodiment of the present invention, a general approach for deconvolution is presented and evaluated based on digital filters derived from linear discriminate analysis (LDA). This approach is based on maximizing the resolution between time coincident and non-coincident events. Simulations of the instrument response were performed to assess the scope of use of the algorithm. Measurements performed using a set of photomultiplier tubers for optical detection of CW sources were performed to experimentally. A highly complex instrument response function was observed and successfully deconvolved. Although a specific example is provided for photon counting, the proposed LDA-based digital filtering may serve as a general approach for deconvolution in other applications as well. In other words, recovery of the impulse response is a means to an end for photon counting, but the scope of use is more general in terms of signal recovery and processing.

Accordingly, presented herein is an approach and assembly for the design of a digital filter for signal deconvolution based on the use of LDA to automatically inform the filter design. The herein disclosed approach differs in that it is based off of maximizing the resolution between detector events directly from experimental data. The signal and the nose are integrated into training sets in the LDA approach, such that the filters are inherently optimized to include impact of noise of the filter, with no prior knowledge or initial guess values. The resulting digital filter can then quickly be applied to streaming data, resulting in deconvoluted impulse-response data amenable to threshold based counting analysis.

Theoretical Foundation:

The digital filter design proposed in this work targets recovery of a temporally sharp impulse for an arbitrary detector impulse response function using a digital filter constructed based on linear discriminate analysis (LDA). LDA is a supervised approach for separating high-dimensional data into distinct classes. In the present application for digital filtering, the length of the filter N defines the dimensionality of the space. Two classes are defined for separation: i) one class in which the voltage transients are time-coincident with the center of the filter; and ii) another in which they are non-coincident, either preceding or following by random time-steps. The co-incident training set is generated from numerous independent measurements of the single-photon response function (e.g., obtained from the dark counts), with each measurement corresponding to a single point in the N-dimensional space of the filter. The non-coincident set was generated by randomly time-shifting either the same or a new co-incident data set. An example of this process for simulated data is shown in FIG. 11. Defining the classes in this manner maximized the resolution between transients initiated at the filter center and all other closely neighboring and temporally overlapping transients.

In the LDA, the variance matrix in the coincident class S_(c) is given by equation (17), in which N_(c) is the number of coincident waveforms used in the training set, V_(c) _(i) is the ith time-dependent coincident single-photon response, and V_(c) is the mean waveform.

$\begin{matrix} {S_{c} = {\frac{1}{N_{c}}{\sum\limits_{i = 1}^{N_{c}}{\left( {V_{c_{i}} - \overset{\_}{V_{c}}} \right)\left( {V_{c_{i}} - \overset{\_}{V_{c}}} \right)^{T}}}}} & (17) \end{matrix}$

The length L of the wafeform spans the temporal range of the single-photon voltage transient. Using an analogous definition for the variance in the non-coincident class S_(n), the within class variance is given by equation (18). S _(w) =S _(c) +S _(n)  (18)

The unit vector F that maximizes the Fischer linear discriminant for a two-class system is generated from S_(w) ⁻¹ according to equation (19).

$\begin{matrix} {F = \frac{S_{w}^{- 1}\left( {\overset{\_}{V_{c}} - \overset{\_}{V_{n}}} \right)}{{S_{w}^{- 1}\left( {\overset{\_}{V_{c}} - \overset{\_}{V_{n}}} \right)}}} & (19) \end{matrix}$

The vector F represents the direction in the L-dimensional space of the filter that maximizes the separation between coincident and non-coincident transients projected onto the axis.

When the center of the filter is time-coincident with the onset of the transient, the scalar output of the filter is given by the dot product of the filter with the transient. Analogously, application of the filter function in a non-coincident position will correspond to a projection of the non-coincident class. Consequently, convolution of the time-trace by the digital filter F generated by LDA maximizes the contrast between coincident and non-coincident events to recover deconvolved δ-function single-pixel maxima at the location of the transient onsets.

Following digital filtering with the LDA-based filter, photon counting of the recovered impulse responses is performed by counting the number of pixels above a selected threshold in the post-filtered deconvolved time-trace. In practice, high-pass filtering is also subsequently performed prior to counting to remove a rolling low-frequency background introduced by the filter. The rolling features arise as a consequence of the maximization of the differential between coincident and non-coincident events by LDA, rather than a maximization of the absolute voltage above baseline.

Simulations:

Simulations with known ground-truth results were performed to assess the performance of LDA-derived digital filtering relative to both conventional photon counting and signal averaging. To produce the training sets, each photon generated a random peak height corresponding to a lognormal distribution in gain in the PMT. A mean μ_(L) of 7 mV and standard deviation σ_(L) of 4 mV describing the lognormal distribution were selected based on measured characteristics for PMTs similar to those used in the present disclosure. The temporal impulse response was also approximated by a lognormal function. For the coincident training set, the initial onset of the pulse was assumed to be at time zero in the center of the filter. The non-coincident training set was generated by offsetting the initial onset by a uniformly distributed shift Δτ over a range spanning the filter length. Each trial also included addition of random 1/f noise, generated by Fourier transformation of normally distributed time-trace, multiplication by √{square root over (1/(f+b))}, in which f is the frequency and b=0.005 is a constant describing the time-scale for 1/f fluctuations, followed by inverse Fourier transformation to recover a time-trace. Qualitatively similar results were observed with normally distributed random noise.

In the simulations of the experimental data acquisition, each time-point contained a random Poisson-distributed number of photons with a mean of λ, where λ was typically less than 1. The time-traces also included 1/f noise calculated identically as the simulated data used to generate the training sets.

The probability of two or more photons arriving within a single time-step also increases as the mean of the Poisson distribution approaches unity. The LDA-filtering approach is designed to maximize the separation between transients produced at separations of more than one time-step of the digitizer, and will therefore not be able to correct for this source of bias. The bias can be corrected by relating the measured counts distributed according to a binomial distribution to the underlying number of photons described by a Poisson distribution. Using algorithms developed previously, the mean λ and the standard deviation σ_(λ) of the Poisson distribution in a given Δt time window are given below, where p is the mean probability for successful counting and N is the number of measured time-points used to assess the mean.

$\begin{matrix} {\lambda = {- {\ln\left( {1 - p} \right)}}} & (20) \\ {\sigma_{\lambda}^{2} = {\frac{p}{N\left( {1 - p} \right)} = \frac{e^{\lambda} - 1}{N}}} & (21) \end{matrix}$

For a sufficiently large number of measurements, the measured mean of p provides a reasonable estimate for the true mean of p, from which the mean and standard deviation of the underlying Poisson distribution can be determined through equations (20) and (21).

Results and Discussion:

Simulations:

Simulations were performed to evaluate the characteristics of the filtering approach. Training sets of 6000 coincident and 6000 non-coincident time-traces were used to generate the digital filter shown in FIG. 11. Three representative images of each of the training set data are also shown in FIG. 11. Still referring to FIG. 11, the histogram shows the distribution of values for the dot products of the training set time-traces with the coincident digital filter, demonstrating the degree of resolution between the two classes.

Application of the filter was performed on a simulated data representing a sensor exhibiting an exponential decay temporal response function (FIG. 12). Referring to FIG. 12, simulated detector data of three measured intensities are shown. In FIG. 12, the temporal location of each pulse is apparent by eye in the simulated raw data, though a threshold counting technique would not be able to successfully count the number of pulses for incidences of significant pulse lineup. After LDA filtering, the temporal location of each pulse is marked by a large positive transient, and can be threshold counted as represented by the threshold and count markers in FIG. 12.

Still referring to FIG. 12, in this simulated data set, the temporal locations of each pulse are known omnipotently, and the quality of the technique can be assessed as a function of threshold position. The ratio of false positives versus true positives is swept across the full range of positive values as shown in FIG. 13 as a receiver operating characteristic curve. Referring to FIG. 13, a data point at the top-left of the graph represents a dataset that can be threshold counted. As the pulse pileup becomes more prominent, the uncertainty of a pulse position increases, though the algorithm still performs well at high intensities and pulse pileups (λ=0.4).

Experimental Data:

The experimental dataset was chosen to be a more challenging temporal response function than a simple exponential decay. The photomultiplier load was terminated with a 330Ω resister in parallel with an internal 1 M Ω resister in an oscilloscope. By using a long transmission cable, a series of reflections arose from each transient, with a ringdown time dictated by the cable capacitance and the terminating resistor (FIG. 14a ). After generation of the LDA filter (FIG. 14b ) and LDA filtering the data (FIG. 14c , labeled “LDA filtered data”), a high pass filter was applied to adjust the shifted baseline (FIG. 14c , labeled “LDA+HP filtered data”). The temporal location of the initial impulse is the position of the first and largest spike in the raw data, which the LDA analysis marked with a strong positive transient. A higher optical intensity dataset is shown in FIG. 15, where the temporal location of each transient is less obvious. The LDA and high pass filtered data (FIG. 15, labeled “LDA+HP filtered data”) predicts the starting position of each transient in the time series with apparently high success.

The temporal location of every transient is not known a priori for the experimental data, so a ROC plot cannot be trivially constructed. However, from Muir et al., it has been determined that the integrated voltage of the time series data is directly proportional to the number of incident photons. For long acquisition times, the expected number of counts can then be cross estimated by this technique (FIG. 16). The LDA filtering tended to recover a better estimator of the number of counts than threshold counting the raw data (which overestimated the counts) or waiting for the response function to time out (which underestimated the number of counts).

Discussion:

The demonstrated LDA method was able to recover impulse responses from both simulated model waveforms in FIGS. 1 and 2, as well as the highly convolved experimental ringing waveforms shown in FIGS. 4 and 5 with low false positive rates. Furthermore, deconvolution could be performed quickly once the appropriate filter function was determined by LDA. However, training of the LDA filter was a relatively computationally expensive step.

The output of the LDA filter produced a rolling background drift over distances comparable to the length of the filter function. The origin of this baseline drift can be understood by considering the nature of the LDA algorithm, in which the filter is designed solely to maximize the separation between coincident and non-coincident classes. The baseline drift is irrelevant to the algorithm, and therefore is not actively corrected by the LDA filter itself. Passing the filtered data through a second high-pass filter removed the baseline offset to make the data more amenable to photon counting based on a simple threshold. Because convolution is associative, the LDA and high-pass filter can ultimately be combined into a single digital filter to further reduce computational costs.

The act of deconvolution resulted in a reduction in the overall signal to noise ratio of the raw measurements, most notably through a reduction in the integrated signal and a significant increase in the variance within the background. Deconvolution retains only the high frequency content within the signal capable of recovering an impulse response, which explains the significant reduction in the integrated area under each transient peak. Furthermore, additional ringing from imperfect impulse recovery was apparent in the experimental dataset.

However, in photon counting applications such as considered herein, both the random noise in the baseline and noise from variance in the integrated areas under the transients are naturally removed through the counting process. As a result, no significant loss in S/N in the measured counts can be expected following deconvolution of photon/ion/electron counting measurements, provided the raw S/N of the measurements is sufficient to recover deconvolved transients still rising above threshold. In fact, significant improvements in the confidence of photon arrival times was observed from analysis of the deconvolved simulated data, as well as increased linear dynamic range through the removal of counting paralysis.

It is worth noting that one LDA algorithm performed reasonably well for two very different incident waveforms. No explicit models were required for treating the measurement noise or the variance in signal intensities. All of these effects were implicitly incorporated into the algorithm through the training set used to produce the LDA filters.

From inspection of FIG. 15, it can be seen that the post-filtered dataset still exhibits residual ringing effects following application of the LDA filter. Some of the ringing excursions rise high enough to appear as false counts, which may explain the slight positive bias arising in FIG. 6 as a result of double-counting. In principle, the post-filtered data could be used to again train a second stage of LDA filtering, and iterate to convergence. Because of the associative properties of convolution, these collective iterations of filter functions can be combined into a single optimized filter.

Although the above described embodiment is focused on photon and ion counting, it is conceivable that the herein described approach may find broader applications in high-speed data transmission. Once it is possible to reliably deconvolve an impulse response, the same filter function should similarly be able to recover deconvolved forms of more complicated functions.

Although only a few examples of this invention have been described in detail above, those skilled in the art will readily appreciate that many modifications are possible without materially departing from the novel teachings and advantages of the invention. Accordingly, all such modifications are intended to be included within the scope of this invention as defined in the following claims. 

The invention claimed is:
 1. A quantitative optical microscopy arrangement, comprising: a light source configured to provide optical energy to a sample region; a detector configured to receive light reflected from or emitted through the sample, the detector comprising: an optical-voltaic converter configured to convert the received light into a representative voltage, wherein the optical-voltaic converter is configured to output the representative voltage as an impulse response; and a digital deconvolution filter operatively connected to the optical-voltaic converter and configured to receive the representative voltage and deconvolve the impulse response, wherein the digital deconvolution filter includes a plurality of coefficients determined based on a linear discriminant analysis of the impulse response of the representative voltage.
 2. The arrangement of claim 1, further comprising: a field programmable gate array coupled to the detector and configured to receive an output of the digital deconvolution filter and provide real-time counting of photons from the sample region. 